Introduction, Analysis and Limiting Behaviour of a Multi-Level Nonhomogeneous Semi-Markov System with a Reducible Embedded Markov Chain

In this paper, we introduce and define the concept of the multi-level nonhomogeneous semi-Markov system. The suggested modelling approach incorporates, for the first time, mobility patterns concerning transitions within or among the departments of an organisation in a semi-Markov context. In this sense, it generalises on the one hand previous Markov departmental models and on the other hand enriches the embedded semi-Markov chain of the system with the characteristic of reducibility. The proposed system is assumed to be open, so, apart from the intra- and inter-departmental mobility, recruitments and departments are also defined. In the first part of the paper, iterative equations are provided for the interval transition probabilities as well as for the system’s structure in order to fully describe the system’s evolution through time. The study of the limiting behaviour of the system follows. Towards this goal, we determine initially the limiting probabilities of the embedded reducible non-homogeneous Markov chain, which allows us in turn to specify the limiting probabilities of the reducible non-homogeneous semi-Markov chain. On the basis of these, the limiting transition probabilities and the limiting population departmental structure of the system are found, under certain conditions. Last, the theoretical results are illustrated numerically with representative data of manpower systems.